Abstracts are presented on a method of numerical solution of the Navier-Stokes equation for the flow about arbitrary airfoils, using a numerically generated curvilinear coordinate system having a coordinate line coincident with the body contour. Results of continuing research are reported and include:  application of the Navier-Stokes solution in the vorticity-stream function formulation to a number of single airfoils at Reynolds numbers up to 2000; programming of the Navier-Stokes solution for multiple airfoils in the primitive variable formulation; testing of the potential flow solution of multiple bodies; and development of a generalized coordinate system program

Thompson, J. F.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19750017802 2020-03-22T21:42:42+00:00Z
i	 i	 I
Semi-Annual Status Report
for the pericd
January 1, 1975 through June 30, 1975
(NASA-CH-143039)	 NI'MER:CAL SOLUTION OF THE	 575-25874
MAVIER-STOKFS EQUATIONS FVF ARBITPAPY
TWO DIMENSIUNAL MULTI-ELEMtHT AIRFOILS
SPaiannual status Report, 1 Jan. - 30 Jun. 	 Unclas
1975 ( Mississippi State Univ., Mississippi 	 G3/02 27321
Submitted Under
NASA Grant NGR-25-001-055
ii
entitled
NUMERICAL SOLUTION OF TIE NAVIER-STOKES EQUATIONS
FOR ARBITRARY TWO-DIMENSIONAL MULTI-ELEMENT AIRFOILS
i
Mississippi State University
Aerophysi.cs and Aerospace Engineering
Mississippi State, MS
	 39762
by
Dr. Joe F. Thompson
^,^0 y^oiirz^^,A
	
h ^	 s
N" = m rte'
t"
 ^
T '"^ ^ CTS
A n CD '^
^r = F ^ cn dti
49 19 b
July 1, 1975
L_
During the reporting peer::od the .following results have been accomplished:
(1) The incompressible Navier-Stokes solution in the vorticity-stream
function formulation has been applied successfully to a number
of single airfoils at Reynolds numbers up to 2000. Initial
progress to higher Reynolds numbers has also been made. Results
were presented in the reports (1), (2), and (3) below. A Ph.D.
dissertation on this work was accepted as noted below and the
degree was granted.
(2) The incompressible Navier-Stokes solution in the primitive
variable formulation (velocity-pressure) has been programmed for
multiple airfoi.ls with general segment arrangements and is now
in operation and under analysis. Initial results were presented
in the report (3) below.
(3) The coordinate system program has been gerneralized to allow any
segment arrangement for multiple bodies, and a number of arrange-
ments have been investigated. Some results were presented in the
report (3) blow.
(4) The potential flow solution for multiple bodies has been tested
by comparison with t:ie analytic solution for a circular cylinder
pair and found to be quite good. Some results were included in
report (3) below.
(S) Prog-amming of the compressible Navier-Stokes solution for single
bodies using :
	 -)rimitive variable formulation has begun.
During the report i.ug period the following conference presentations
were made (abstracts attached):
(1) "Numerical Solution of the Navier-Stokes Equations for Arbitrary
Two-Dimensional Airfoils," F. C. Thames, Joe F. Thompson, and
C. W. Mastin, Proceedings of NASA Conference on Aerodynamtc
Anblyse_: :requiring Advanced Computers, Langley Research Center,
Hampton, Va., March 4-6, 1975. To be published as NASA-SP 31,7.
(2) "Numerical Solutions of the Unsteady Navier-Stokes Equations for
,trbitrary Bodies Using Boundary-Fitted Curvilinear Coordinates,"
J. F. Thompson, F. C. Thames, R. L. Walker, and S. P. Shanks,
Proceedings of Arizona/AFOSR Symposium on Unsteady Aerodynamics,
Univ. of Arizona, Tucson, Ariz., March 18-20, 1975.
(3) "Use of Numerically Generated Body-Fitted Coordinate Systems for
Solutions of the Navier-Stokes equations," J. F. Thompson, F. C.
Thames, C. W. Mastin, and S. P. Shanks, Proceedings of AIAA 2nd
Computational Fluid Dynamics Conference, Hartford, Conn., June
19-20, 1975.
During the reporting period the foll--sing Ph.D. dissertation was
accepted ',abstract attached):
"Numerical Solution of the Tneompressible Navier-Stokes Equations
abort Arbitrary Two-Dimensional Bodies," Frank C. Thames, Ph.D.
Dissertation, Mississippi State University, May 1975.
NUMERICAL SOE.U'r T ' N OF THE NAVIER-STOKES EQUATIONS
FOR ARBIT ....Y TWO-DIMENSIONAL AIRFOILS
Joe F. Thompson and Frank C. Thames
Department of Acrophysics and Aerospace Engineering
Mississippi State University, Mississippi State, MS	 39762
Research Spr,nsored by NASA Langley Research Center, Contract NGR 25-001-055
Proceedings of NASA Confers—ce on Aerodynamic Analyses Requiring Advanced
Computers, Langley Research Center, Hampton, Va., March 4-6, 1975
ABSTRACT
A method of numerical sol:tion of the Navier-Stokes equations for the
flow about arbitrary airfoils, or other bodies, is presented. This meth^A
utilizes a numerically generated curvilinearroo:-dinate sys=em having a
coordinate line coincident with the body contour. Streamline, velocity
profiles, and pressure and force coefficients for several airfoils and an
arbitrary rock are given. Potential flow solutions are also presented.
The procedure is also capable of treating multi-element airfoils, and
potential flow results are presented therefor.
NUMERICAL SOLUTIONS OF Till UNSTEADY NAVIER-STOKES EQUATIONS
FOR ARBITkARY BODIES USING
BOUNDARY-F1TT-D CURVILINEAR COORDINATES
Joe F. Thompson, Frar.., C. Thamen
Ray L. Walker, Samuel P. Shanks
Department of Aerophysics and Aerospace Engineering
Mississippi State University
Research sponsored by Langley Research Center, NASA, Grant NCR 25-001-055
and Office of Naval Research, USN, Contract N00014-74-C-0373-P0001
Proceedings of Arizona/AFOSR Syu,,)osium on Unsteady Aerodynamics, Univ. of
Arizona, Tucson, Ariz., March 18-20, 1975
ABSTRACT
A numerical solution of • ' , e time-dependent, two-dimensional incompres-
sible Navier-Stokes equations that can treat the unsteady laminar flow
about bodies of arbitrary shape, such as airfoils, as naturally as simple
bodies has been developed. Unsteady boundaries, such as deforming bodies
or frc? surfaces, can also be treated. This solution is based on a method
of aute iatic numerical generation of a general curvilinear coordinate system
with coo dinate lines coincident with all boundaries of a general multi-
connected egion containing any number of arbitrarily shaped bodies. The
curvilinear -iordinates are generated as the solution of two elliptic partial
differential e, ations with Dirichlet boundary conditions, one coordinate
being specified L be constant on each of the boundaries, and a distri-
bution of the other being specified along the boundaries. No restrictions
are placed on the shape of the boundaries, v?hich may even be time-dependent,
and the method is not restricted to two dimensions or single bodies.
Coordinate lines may be concentrated as desired along the boundaries.
Spacing of the coordinate lir^s encircling the body may be controlled by
adjusting parameters in the partial differential equations for the coordinates.
i
USE OF NUMERICALLY GENERATED BODY-FITTED COORDINATE SYSTEMS
FOR SOLUTION OF THE NAVIER-STOKES EQUATIONS
J. F. Thompson, F. C. Thames, C. W. Mastin, and S. P. Shanks
Department of Acrophysics and Aerospace Engineering
Mississippi State University, Missi.sippi State, MS	 39762
Research sponsored by Langley Research Center, NASA, Grant NCR 25-001-055
and Office of Naval Research, USN, Contract N00014-74-C-0373-P0001
1'roceedi ►.gs of AIAA 2nd Computational Fluid Dynamics Conference, Hartford,
Conn., June 19-20, 1975
i	 ABSTRACT
These coordinate systems remove geometric considerations from numerical
solutions of the Navier-Stokes equations so that the flow about bodies of
arbitrary shape can be treated. Fields with multiple bodies or with time-
dependent boundaries may also be treated. Several types of such systems
and methods of control of the coordinate line spacing and rientation are
discussed. hit computations are done on a fixed rectangular field with
square n,esh without any interpolation regardless of the shape or movement
of the ph ysical boundaries and regardless of the spacing ^,, he coordinate
lines. Potential and viscous flow solutions are presente, 	 -r airfoils
with attached and separated flaps and for hydrofoils near a free surface.
L J	 I	 I	 i
ABSTRACT
Frank C. Thames, Doctor of Philosophy, 1975
Major: Engineering, Department of Aerophysics and Aerospace Engineering
Title of Dissertation: ),:umerical Solution of the Incomprv!,sible N avier-
Stokes Equations About Arbitrary Two-Dimensional
Bodies
Directed By: Dr. Joe F. Thompson
Pages in Dissertation:	 205	 Words in Abstra..: 515
ABSTRACT
The purpose of this research was to develop methods capable of
producing numerical F,olutions of the two-dimensional, incompressible,
time-dependent Navier-Stokes equations about arbitrary two-dimensional
bodies. As a preliminary step a method of automatic numerical generation
of a gereral curvilinear coordinate system having a constant coordinate
line coincident with each boundary of a general multi-connected physical
flow region was developed. These curvilinear coordinates are generated
as the solution cf ta ro elliptic partial differential equations with
Dirichlet boundary conditions. Onc natural coordinate is specified to
be constant along each boundary of the physical region while a distri-
bution of the other is specified along the boundary contours. "he trans-
formlation is initiated by the selection of a generating elliptic partial
differential system of which the natural coordinates are a soll_A on in
the physical flow plane. This pe neraLing system is transformed by inter-
9
changing the dependent and independent variables producing a set of two
coupled, quasi-linear elliptic equations for the cartesian coordinates
as functions of the natural curvilinear coordinates. If tike original
generating elliptic system obeys a maxlrnum principle and if the range
of the r, tural coordinate which varie, along; the physical contours is
made the same on each boundary, then `he transformed plane will be
rectangular. The quasi-linear transformation equations are solved for
the rhysical coordinates in the transformed plane using SOR techniques.
The boundaries of the physical region appear simply as input boundary
conditions for the numerical solution in the transformed plane. No
restrictions are placed on the shape of the boundaries in the physical
plane which may even be time dependent. Further, the method is not
restricted t) two dimensions. Coordinate lines may be concentrated as
desired along the physical contours while control over the spacing of
the radial coordinate is adjusted by varying the generating elliptic
system.
Once the natural coordinates are generated for a given physical
domain, the vorticity-stream function formulation of the Navier-Stokes
equations are transformed to the rectangular transformed plane. The
equations are then approximated using central differences for the space
derivatives and a backward first-order time difference for the vorticity
time derivative. The flow solution is carried out by an implicit
method with SOR iteration being used to converge the elliptic space
variation at each time step. The vorticity distribution on the body
surface is calculated at cacti time step utilizing; a modified multi-
dimensional false position iteration designed to force the tangential
velocity component oil 	 body surface to zero. Dirichlet conditions
correspondin g to irrotational flow were used at the remote boundary.
Successful solutions about five different bodies were generated. These
included a circular cylinder, cambered and flapped Karman-Trefftz air-
foils, a O'Ittingen 625 airfoil, and a general arbitrary t,ody-denoted the
cambered rock. Reynolds nurnbers for the solut i ons varied from 200 to
2,000 at angles of attack from zero to fifteen degrees. Data defining
stream function and vorticity contours, boundary layer velocity profiles,
pressure dirtributions, and force coefficients are presented for all
so' tons. A mildly successful solution about a NACA 0016 airfoil at a
Reynolds number of 10,000 was also accorrp11stied.


Numerical solution of the Navier-Stokes equations for arbitrary two-dimensional multi-element airfoils

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server